Limiting Behaviors of High Dimensional Stochastic Spin Ensembles

Journal Article

Lattice spin models in statistical physics are used to understand magnetism.
Their Hamiltonians are a discrete form of a version of a Dirichlet energy,
signifying a relationship to the Harmonic map heat flow equation. The Gibbs
distribution, defined with this Hamiltonian, is used in the Metropolis-Hastings
(M-H) algorithm to generate dynamics tending towards an equilibrium state. In
the limiting situation when the inverse temperature is large, we establish the
relationship between the discrete M-H dynamics and the continuous Harmonic map
heat flow associated with the Hamiltonian. We show the convergence of the M-H
dynamics to the Harmonic map heat flow equation in two steps: First, with fixed
lattice size and proper choice of proposal size in one M-H step, the M-H
dynamics acts as gradient descent and will be shown to converge to a system of
Langevin stochastic differential equations (SDE). Second, with proper scaling
of the inverse temperature in the Gibbs distribution and taking the lattice
size to infinity, it will be shown that this SDE system converges to the
deterministic Harmonic map heat flow equation. Our results are not unexpected,
but show remarkable connections between the M-H steps and the SDE Stratonovich
formulation, as well as reveal trajectory-wise out of equilibrium dynamics to
be related to a canonical PDE system with geometric constraints.